Geometric invariant theory for geometers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:15:34Z http://mathoverflow.net/feeds/question/103784 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103784/geometric-invariant-theory-for-geometers Geometric invariant theory for geometers berl13 2012-08-02T14:17:14Z 2012-08-07T18:34:07Z <p>I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry. </p> <p>So my question is if there is any nice reference where people explain geometric invariant theory from a geometric viewpoint. In particular, I am looking for a good reference where the analogies between algebraic geometry and differential geometry are pointed out.</p> http://mathoverflow.net/questions/103784/geometric-invariant-theory-for-geometers/103800#103800 Answer by Eugene Lerman for Geometric invariant theory for geometers Eugene Lerman 2012-08-02T17:19:02Z 2012-08-07T18:34:07Z <p>I would recommend a look at chapter 8 of the <strong>third</strong> edition of <em>Geometric invariant theory</em> by Mumford, Forgarty and Kirwan. It describes a connection between GIT and Hamiltonian group actions in symplectic geometry. </p> <p>(edit) You may also like <a href="http://arxiv.org/abs/0912.1132" rel="nofollow">Moment maps and geometric invariant theory</a> by Chris Woodward.</p> http://mathoverflow.net/questions/103784/geometric-invariant-theory-for-geometers/103808#103808 Answer by Jason Polak for Geometric invariant theory for geometers Jason Polak 2012-08-02T18:43:31Z 2012-08-02T18:43:31Z <p>If you just want to get a feeling for invariant theory, here are some books that aren't necessarily comprehensive but nevertheless are enlightening at a more leisurely pace as compared to GIT, which would be useful for someone who isn't as familiar with algebraic groups and algebraic geometry:</p> <ul> <li><p>Santos and Rittatore - Actions and Invariants of Algebraic Groups: Minimal prerequisites. A very gentle introduction to some aspects of invariant theory, including some motivation via Hilbert's 14th problem. This book also contains most of the required theory of linear algebraic groups.</p></li> <li><p>Dolgachev - Lectures on Invariant Theory: This takes a more geometric viewpoint and might be something you are interested in. This only requires some basic knowledge of algebraic geometry.</p></li> <li><p>Schmitt - Geometric Invariant Theory and Decorated Principal Bundles: this might also be interesting if you are interested in the geometric applications and the related geometry, though I haven't looked into this book very much, but Part 1 does contain a fairly leisurely-looking introduction to GIT</p></li> </ul> <p>There is also Popov's and Vinberg's treatise "Invariant Theory" in the Ecyclopedia of Mathematical Sciences Volume 55 (Springer) which contains a good summary of the classical results in characteristic zero.</p> http://mathoverflow.net/questions/103784/geometric-invariant-theory-for-geometers/103855#103855 Answer by Buschi Sergio for Geometric invariant theory for geometers Buschi Sergio 2012-08-03T11:02:33Z 2012-08-03T16:02:54Z <p>For a "more classical" point of view:</p> <p>"An introduction to Invariants and Moduli " by S. Mukai.</p> <p>Fogarty J. "Invariant theory" (Benjamin, 1969)</p> <p>For a introduction to Mumford's:</p> <p>P. E. Newstead, "Introduction to Moduli Problems and Orbit Spaces" </p> <p>Anyway you have to learn (before or after) a Gothendieck-categorical background:</p> <p>Fondements de la géométrie algébrique (Grothendick). The Hilbert schema chapter is very important (need the Hartshorne "Algebraic Geometry" as base)</p> <p>Or in more gentle way: Fundamental Algebraic Geometry. Grothendieck's FGA Explained - Fantechi B., Göttsche, L., Illusie L. </p> http://mathoverflow.net/questions/103784/geometric-invariant-theory-for-geometers/104064#104064 Answer by kalafat for Geometric invariant theory for geometers kalafat 2012-08-06T00:17:56Z 2012-08-06T00:17:56Z <p>Read the survey in my <a href="http://front.math.ucdavis.edu/0611.5771" rel="nofollow">article</a> and go over the references therein. It is written with exactly similar intentions you have asked for.</p>